Barycentric Markov processes and stability of stochastic integrators

Abstract: This thesis consists of four papers that broadly concerns two dierent topics. The rsttopic is so-called barycentric Markov processes. By a barycentric Markov process wemean a process that consists of a point/particle system evolving in (discrete) time,whose evolution depends in some way on the mean value of the current points in thesystem. In common for the three rst papers which are on this topic is that we studyhow all the points of the so-called core (a certain subset of points in the system) of thesystem converge to the same point.The rst article concerns how an N-point system behaves when we reject the K < N=2points that minimize the sample variance of the remaining N ?K points (the core). Wethen replace the rejected points with K new points which follow some xed distributionand which are all independent from the past points. When K = 1 this is equivalent torejecting the point which is furthest from the center of mass. We prove that under ratherweak assumptions on the sampling distribution, the points of the core converge to thesame point as well as that regardless of any assumptions on our sampling distribution,the sampling variance of the core converges to zero or the core "drifts o to innity".The second article concerns a similar problem as the rst one. We once again consideran N-point system but at each time step we reject the point furthest from the centerof mass multiplied by a positive number p and replace it with a point from a xeddistribution with full support on [0; 1], which is independent from all past points. Ifp = 1 we obtain a special case of the previous article. If p 6= 1 it turns out thatthis process behaves very dierently from the process in the rst article, the stationarydistribution to which the core points converge turns out to be a Bernoulli distribution.The third article studies yet another N-point system but now on a discrete circle.During each time step we compute the distances for each point from the mean of it'stwo neighbours and reject the one with largest such distance (thereby obtaining ourcore) and replace it with a new point independent from past points. Two dierentcases are considered, the rst is with uniformly distributed points in [0; 1] and the otheris with a discrete uniform distribution (i.e. uniformly distributed on an equally spacedgrid).The fourth and last article is on the topic of stochastic calculus. The main objective isto study "stability" of integrators for stochastic integrals. We examine how converging sequences of processes in the role of integrators retain their convergence propertiesfor their corresponding integrals when the integrators are transformed under certainclasses of functions. The convergence is on one hand in the uniform (over compact timeintervals) in Lp-sense and on the other hand in the UCP-sense (uniform convergence inprobability on compact time intervals). We examine processes with quadratic variations(along some rening sequence) transformed by absolutely continuous functions as wellas Dirichlet processes transformed by C1 functions.